Attention to detail: When he ordered a bespoke suit from Anderson and Shepherd in Savile Row, the carpet was rolled back and Astaire danced on the parquet floor to check that the fit of his coat never came away from his collar.
More from Mike Brown, (any links gratefully received):
Astaire was famous for practicing his steps over and over until he didn't have to think about them. I read somewhere that his test was, while dancing, to read a book. If he could read the book and retain what he read, then he knew he had learned the dance.
Also, he and his choreographers never wrote anything down, so they worked off of muscle memory.
Last anecdote: reputedly, he was asked by a young performer how to polish his act. Astaire said to get the act as perfect as you can, then cut two minutes.
More How we work.